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The Shooting Method for the Solution of One-Dimensional BVPs

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Boundary Value Problems for Engineers

Abstract

The shooting method transforms the boundary-value ODE into a system of first order ODEs, which can be solved by the initial-value methods. The boundary conditions on one side of the given interval is used as initial conditions. The additional initial conditions needed are assumed, the initial-value problem is solved, and the solution at the other boundary is compared to the known boundary conditions on that boundary. The initial conditions guessed on one boundary is varied iteratively until the boundary conditions on the other boundary are satisfied.

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Notes

  1. 1.

    For refreshing information on the structure of RK4, Euler and Heun’s methods and their MATLAB applications, readers may refer to textbooks on numerical methods, such as [4,5,6,7].

  2. 2.

    It is particularly advantageous to employ dsolve function here, since the analytical solution obtained using this function is useful in assessing the accuracy of the solutions obtained with the (approximate) numerical methods.

  3. 3.

    Apparently, analytical solution is easy to obtain and more feasible for that linear, homogeneous ODE.

  4. 4.

    The study of “nonlinear methods of shooting” brings more insight to the study of linear Shooting techniques, helping to devise more general algorithms for BVP solvers.

  5. 5.

    This method is not thoroughly understood as is evidenced by the fact that there is no generally accepted rule for picking the nodes [37]. However, some theoretical results and available strong heuristic motivations are useful in selecting the shooting points.

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  1. Department of Biomedical Engineering, Yeditepe University, Istanbul, Turkey

    Ali Ümit Keskin

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Keskin, A.Ü. (2019). The Shooting Method for the Solution of One-Dimensional BVPs. In: Boundary Value Problems for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-21080-9_5

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  • DOI: https://doi.org/10.1007/978-3-030-21080-9_5

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  • Print ISBN: 978-3-030-21079-3

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